Which term is correct, line or edge, when connecting non-adjacent vertices? I understand that lines that connects two adjacent vertices of a regular polygon are edges but is the same term used for lines that connect non-adjacent vertices? For example, the link below is of an octagon with every pair of vertices connected and my question is about the interior lines that intersect with one another. I just want to make sure I use the correct term when referencing them. Thanks in advance!
https://commons.wikimedia.org/wiki/File:7-simplex_t0.svg
 A: The appropriate term is "diagonal".
You can even distinguish "first diagonal", "second diagonal", etc, depending upon how many vertices are skipped. (Under this naming convention, an edge is sometimes called a "zeroth diagonal".)
A: A line in mathematics is a 1-dimensional straight infinite fabrics. Even so it sometimes also is being used for line segments, then silently dropping the last bit. OTOH a line in common sense also is what a pencil draws on a sheet of paper. Thus there a line neither will be infinite nor has to be straight.
Wrt. polygons the 1-dimensional bounding bits usually are called sides. But then wrt. to a cube say the side will be 2-dimensional. Thus that very term really is meant to have co-dimension 1. - The lines (in the drawing sense) running more or less diagonally through a polygon, thereby connecting non-neighbouring vertices, usually are called chords.
Edges OTOH usually are the 1-dimensional bounding elements of polyhedra. As such this very term also is being taken over for higher-dimensional polytopes. But in common sense the term edge might demark the rim line of a cliff, thus demarking the margin between the more or less horizontal and vertical planes, or even the border between 2 countries (thus considered as the connection between 2 bounded planes). So it is rather being used within polyhedral complexes, not in context of a single polygon.
--- rk
